Chapter 13 Supplement
Linear Programming
Operations Management - 5th Edition
Roberta Russell & Bernard W. Taylor, III
Copyright 2006 John Wiley & Sons, Inc.
Beni Asllani
University of Tennessee at Chattanooga
Lecture Outline





Model Formulation
Graphical Solution Method
Linear Programming Model
Solution
Solving Linear Programming Problems
with Excel
 Sensitivity Analysis
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-2
Linear Programming (LP)
A model consisting of linear relationships
representing a firm’s objective and resource
constraints
LP is a mathematical modeling technique used to
determine a level of operational activity in order to
achieve an objective, subject to restrictions called
constraints
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-3
Types of LP
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-4
Types of LP (cont.)
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-5
Types of LP (cont.)
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-6
LP Model Formulation
 Decision variables

mathematical symbols representing levels of activity of an
operation
 Objective function



a linear relationship reflecting the objective of an operation
most frequent objective of business firms is to maximize profit
most frequent objective of individual operational units (such as
a production or packaging department) is to minimize cost
 Constraint

a linear relationship representing a restriction on decision
making
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-7
LP Model Formulation (cont.)
Max/min
z = c1x1 + c2x2 + ... + cnxn
subject to:
a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
:
am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-8
LP Model: Example
RESOURCE REQUIREMENTS
PRODUCT
Bowl
Mug
Labor
(hr/unit)
1
2
Clay
(lb/unit)
4
3
Revenue
($/unit)
40
50
There are 40 hours of labor and 120 pounds of clay
available each day
Decision variables
x1 = number of bowls to produce
x2 = number of mugs to produce
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-9
LP Formulation: Example
Maximize Z = $40 x1 + 50 x2
Subject to
x1 + 2x2 40 hr (labor constraint)
4x1 + 3x2 120 lb (clay constraint)
x1 , x2 0
Solution is x1 = 24 bowls
Revenue = $1,360
Copyright 2006 John Wiley & Sons, Inc.
x2 = 8 mugs
Supplement 13-10
Graphical Solution Method
1. Plot model constraint on a set of coordinates
in a plane
2. Identify the feasible solution space on the
graph where all constraints are satisfied
simultaneously
3. Plot objective function to find the point on
boundary of this space that maximizes (or
minimizes) value of objective function
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-11
Graphical Solution: Example
x2
50 –
40 –
4 x1 + 3 x2 120 lb
30 –
Area common to
both constraints
20 –
10 –
0–
x1 + 2 x2 40 hr
|
10
|
20
Copyright 2006 John Wiley & Sons, Inc.
|
30
|
40
|
50
|
60
x1
Supplement 13-12
Computing Optimal Values
x1 + 2x2 = 40
4x1 + 3x2 = 120
x2
40 –
4 x1 + 3 x2 120 lb
4x1 + 8x2 = 160
-4x1 - 3x2 = -120
30 –
20 –
x1 + 2 x2 40 hr
5x2 =
x2 =
40
8
x1 + 2(8) =
x1
=
40
24
10 –
8
0–
|
10
| 24 |
20
30
| x1
40
Z = $50(24) + $50(8) = $1,360
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-13
Extreme Corner Points
x1 = 0 bowls
x2 = 20 mugs
Z = $1,000
x2
40 –
30 –
20 – A
10 –
0–
x1 = 224 bowls
x2 = 8 mugs
x1 = 30 bowls
Z = $1,360
x2 = 0 mugs
Z = $1,200
B
|
10
|
20
| C|
30 40
Copyright 2006 John Wiley & Sons, Inc.
x1
Supplement 13-14
Objective Function
x2
40 –
4x1 + 3x2 120 lb
30 –
Z = 70x1 + 20x2
Optimal point:
x1 = 30 bowls
x2 = 0 mugs
Z = $2,100
20 –A
10 –
0–
B
|
10
|
20
Copyright 2006 John Wiley & Sons, Inc.
| C
30
x1 + 2x2 40 hr
|
40
x1
Supplement 13-15
Minimization Problem
CHEMICAL CONTRIBUTION
Brand
Nitrogen (lb/bag)
Phosphate (lb/bag)
2
4
4
3
Gro-plus
Crop-fast
Minimize Z = $6x1 + $3x2
subject to
2x1 + 4x2  16 lb of nitrogen
4x1 + 3x2  24 lb of phosphate
x 1, x 2  0
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-16
Graphical Solution
x2
14 –
x1 = 0 bags of Gro-plus
12 – x = 8 bags of Crop-fast
2
Z = $24
10 –
8–A
Z = 6x1 + 3x2
6–
4–
B
2–
0–
|
2
|
4
Copyright 2006 John Wiley & Sons, Inc.
|
6
|
8
C
|
10
|
12
|
14
x1
Supplement 13-17
Simplex Method
 A mathematical procedure for solving linear programming
problems according to a set of steps
 Slack variables added to ≤ constraints to represent unused
resources


x1 + 2x2 + s1 = 40 hours of labor
4x1 + 3x2 + s2 = 120 lb of clay
 Surplus variables subtracted from ≥ constraints to represent
excess above resource requirement. For example


2x1 + 4x2 ≥ 16 is transformed into
2x1 + 4x2 - s1 = 16
 Slack/surplus variables have a 0 coefficient in the objective
function

Z = $40x1 + $50x2 + 0s1 + 0s2
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-18
Solution
Points with
Slack
Variables
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-19
Solution
Points with
Surplus
Variables
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-20
Solving LP Problems with Excel
Click on “Tools”
to invoke “Solver.”
Objective function
=E6-F6
=E7-F7
=C6*B10+D6*B11
=C7*B10+D7*B11
Decision variables – bowls
(x1)=B10; mugs (x2)=B11
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-21
Solving LP Problems with Excel
(cont.)
After all parameters and constraints
have been input, click on “Solve.”
Objective function
Decision variables
C6*B10+D6*B11≤40
C7*B10+D7*B11≤120
Click on “Add” to
insert constraints
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-22
Solving LP Problems with Excel
(cont.)
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-23
Sensitivity Analysis
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-24
Sensitivity Range for Labor
Hours
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-25
Sensitivity Range for Bowls
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-26
Copyright 2006 John Wiley & Sons, Inc.
All rights reserved. Reproduction or translation of this work beyond that
permitted in section 117 of the 1976 United States Copyright Act without
express permission of the copyright owner is unlawful. Request for further
information should be addressed to the Permission Department, John Wiley &
Sons, Inc. The purchaser may make back-up copies for his/her own use only and
not for distribution or resale. The Publisher assumes no responsibility for
errors, omissions, or damages caused by the use of these programs or from the
use of the information herein.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-27